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arXiv:0801.0831 · Physical Review A

Graphical nonbinary quantum error-correcting codes

Dan Mei Hu, Weidong Tang, Meisheng Zhao, Qing Chen, Sixia Yu, C. H. Oh

Published Jul 3, 2008• 75 citations

Abstract

In this paper, we present a systematic way based on the nonbinary graph state of constructing good nonbinary quantum codes, both additive and nonadditive, for systems with integer dimensions. With a computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely, ${[[6,2,3]]}_{p}$, ${[[7,3,3]]}_{p}$, ${[[8,2,4]]}_{p}$, and ${[[8,4,3]]}_{p}$ for any odd dimension $\mathit{p}$ and a family of nonadditive codes ${((5,p,3))}_{p}$ for arbitrary $p>3$. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes ${((6,2{p}^{2},3))}_{2p}$ for odd $\mathit{p}$, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.

DOI: 10.1103/physreva.78.012306

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In this paper, we present a systematic way based on the nonbinary graph state of constructing good nonbinary quantum codes, both additive and nonadditive, for systems with integer dimensions. With a computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely,

{[[6,2,3]]}_{p}

,

{[[7,3,3]]}_{p}

,

{[[8,2,4]]}_{p}

, and

{[[8,4,3]]}_{p}

for any odd dimension

\mathit{p}

and a family of nonadditive codes

{((5,p,3))}_{p}

for arbitrary

p>3

. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes

{((6,2{p}^{2},3))}_{2p}

for odd

\mathit{p}

, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.

Graph states are generalized from qubits to collections of

n

qudits of arbitrary dimension

D

, and simple graphical methods are used to construct both additive and nonadditive, as well as degenerate and nondegenerate, quantum-error-correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large

n

and

D

are constructed using simple graphs, except when

n

is odd and

D

is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general

D

, and shown to provide a dual representation of an additive graph code.

We introduce a purely graph-theoretical object, namely the coding clique, to construct quantum errorcorrecting codes. Almost all quantum codes constructed so far are stabilizer (additive) codes and the construction of nonadditive codes, which are potentially more efficient, is not as well understood as that of stabilizer codes. Our graphical approach provides a unified and classical way to construct both stabilizer and nonadditive codes. In particular we have explicitly constructed the optimal ((10,24,3)) code and a family of 1-error detecting nonadditive codes with the highest encoding rate so far. In the case of stabilizer codes a thorough search becomes tangible and we have classified all the extremal stabilizer codes up to 8 qubits.

The codeword-stabilized (CWS) quantum code formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes [see A. Cross et al., e-print arXiv:0708.1021], but only for binary states. Here we generalize the CWS framework to the nonbinary case (of both prime and nonprime dimensions) and map the search for nonbinary quantum codes to a corresponding search problem for classical nonbinary codes with specific error patterns. We show that while the additivity properties of nonbinary CWS codes are similar to the binary case, the structural properties of the nonbinary codes differ substantially from the binary case, even for prime dimensions. In particular, we identify specific structure patterns of stabilizer groups, based on which efficient constructions might be possible for codes that encode more dimensions than any stabilizer codes of the same length and distance; similar methods cannot be applied in the binary case. Understanding of these structural properties can help prune the search space and facilitate the identification of good nonbinary CWS codes.

We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.